54.4.61 problem 1519

Internal problem ID [12783]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1519
Date solved : Friday, October 03, 2025 at 03:47:24 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} \left (x +3\right ) x^{2} y^{\prime \prime \prime }-3 x \left (x +2\right ) y^{\prime \prime }+6 \left (x +1\right ) y^{\prime }-6 y&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 19

ode:=(x+3)*x^2*diff(diff(diff(y(x),x),x),x)-3*x*(x+2)*diff(diff(y(x),x),x)+6*(1+x)*diff(y(x),x)-6*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_2 \,x^{3}+c_1 \,x^{2}+c_3 x +c_3 \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 58

ode=-6*y[x] + 6*(1 + x)*D[y[x],x] - 3*x*(2 + x)*D[y[x],{x,2}] + x^2*(3 + x)*Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{8} \left (2 c_1 \left (x^3-3 x^2+3 x+3\right )-(x-1) \left (4 c_2 \left (x^2-2 x-1\right )+c_3 \left (-3 x^2+2 x+1\right )\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x + 3)*Derivative(y(x), (x, 3)) - 3*x*(x + 2)*Derivative(y(x), (x, 2)) + (6*x + 6)*Derivative(y(x), x) - 6*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**3*Derivative(y(x), (x, 3))/6 + x**2*Derivative(y(x), (x, 2))/2 - x**2*Derivative(y(x), (x, 3))/2 + x*Derivative(y(x), (x, 2)) + y(x))/(x + 1) cannot be solved by the factorable group method